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Mathematical Proofs

Strongly Recommended Prerequisites

Recommended Prerequisites

Last Updated: 8/29/2021

A mathematical proof is a rigorous argument based on straightforward logical rules that is used to convince other mathematicians (including the proof's author) that a statement is true. Any mathematical subject in data science will employ proofs, and the ability to write convincing proofs is an important mathematical skill for data scientists. Unfortunately, learning to write good proofs is not easy. Most mathematicians improve their proof-writing ability by receiving feedback from teachers over the course of their education; it's hard to do that with just a textbook. However, the basic components and principals of writing proofs can be learned from a book. The onus is then on the autodidact to improve their skills by reading proofs and by comparing their own proofs to those of others. Alas, when official solutions are available to textbook exercises it is often the proof solutions that are omitted. However, since most of the books on this website are relatively popular, it is often possible to find unofficial proofs written by others on the internet. Quality varies, of course.

Recommended Books

  1. Mathematical Proofs: A Transition to Advanced Mathematics

    Gary Chartrand, Albert D. Polimeni, Ping Zhang

    Check it out on Amazon!

    Key Features

    Key Topics

    • Counterexamples
    • Direct Proof
    • Equivalence Relations
    • Logical Equivalence
    • Logical Statements
    • Mathematical Induction
    • Mathematical Writing
    • Proof by Contradiction
    • Proof by Contrapositive
    • Proofs in Various Mathematical Domains
    • Sets


    This is a very useful book for learning about proof-writing. You'll learn enough mathematics to have something to prove, as well as the common techniques for writing proofs (direct proof, proof by contradiction, induction, etc.). We recommend reading the first half of the book and doing many of the odd-numbered exercises (for which there are solutions). You can then either proceed to the second half, or return to this book as you cover the material to which the second half relates. It is important that you carefully compare your solutions to the book solutions. A minor deficiency in a proof can spoil the whole argument.